8,451 research outputs found
The Short Path Algorithm Applied to a Toy Model
We numerically investigate the performance of the short path optimization
algorithm on a toy problem, with the potential chosen to depend only on the
total Hamming weight to allow simulation of larger systems. We consider classes
of potentials with multiple minima which cause the adiabatic algorithm to
experience difficulties with small gaps. The numerical investigation allows us
to consider a broader range of parameters than was studied in previous rigorous
work on the short path algorithm, and to show that the algorithm can continue
to lead to speedups for more general objective functions than those considered
before. We find in many cases a polynomial speedup over Grover search. We
present a heuristic analytic treatment of choices of these parameters and of
scaling of phase transitions in this model.Comment: 11 pages, 9 figures; v2 final version published in Quantu
Random Weighting, Asymptotic Counting, and Inverse Isoperimetry
For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of
X and let Gamma(X,mu) be the expected maximum weight of a subset from X when
the weights of 1,...,n are chosen independently at random from a symmetric
probability distribution mu on R. We consider the inverse isoperimetric problem
of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove
that the optimal choice of mu is the logistic distribution, in which case
Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X|
grows. Since in many important cases Gamma(X,mu) can be easily computed, we
obtain computationally efficient approximation algorithms for a variety of
counting problems. Given mu, we describe families X of a given cardinality with
the minimum value of Gamma(X,mu), thus extending and sharpening various
isoperimetric inequalities in the Boolean cube.Comment: The revision contains a new isoperimetric theorem, some other
improvements and extensions; 29 pages, 1 figur
Blind MultiChannel Identification and Equalization for Dereverberation and Noise Reduction based on Convolutive Transfer Function
This paper addresses the problems of blind channel identification and
multichannel equalization for speech dereverberation and noise reduction. The
time-domain cross-relation method is not suitable for blind room impulse
response identification, due to the near-common zeros of the long impulse
responses. We extend the cross-relation method to the short-time Fourier
transform (STFT) domain, in which the time-domain impulse responses are
approximately represented by the convolutive transfer functions (CTFs) with
much less coefficients. The CTFs suffer from the common zeros caused by the
oversampled STFT. We propose to identify CTFs based on the STFT with the
oversampled signals and the critical sampled CTFs, which is a good compromise
between the frequency aliasing of the signals and the common zeros problem of
CTFs. In addition, a normalization of the CTFs is proposed to remove the gain
ambiguity across sub-bands. In the STFT domain, the identified CTFs is used for
multichannel equalization, in which the sparsity of speech signals is
exploited. We propose to perform inverse filtering by minimizing the
-norm of the source signal with the relaxed -norm fitting error
between the micophone signals and the convolution of the estimated source
signal and the CTFs used as a constraint. This method is advantageous in that
the noise can be reduced by relaxing the -norm to a tolerance
corresponding to the noise power, and the tolerance can be automatically set.
The experiments confirm the efficiency of the proposed method even under
conditions with high reverberation levels and intense noise.Comment: 13 pages, 5 figures, 5 table
Convex Graph Invariant Relaxations For Graph Edit Distance
The edit distance between two graphs is a widely used measure of similarity
that evaluates the smallest number of vertex and edge deletions/insertions
required to transform one graph to another. It is NP-hard to compute in
general, and a large number of heuristics have been proposed for approximating
this quantity. With few exceptions, these methods generally provide upper
bounds on the edit distance between two graphs. In this paper, we propose a new
family of computationally tractable convex relaxations for obtaining lower
bounds on graph edit distance. These relaxations can be tailored to the
structural properties of the particular graphs via convex graph invariants.
Specific examples that we highlight in this paper include constraints on the
graph spectrum as well as (tractable approximations of) the stability number
and the maximum-cut values of graphs. We prove under suitable conditions that
our relaxations are tight (i.e., exactly compute the graph edit distance) when
one of the graphs consists of few eigenvalues. We also validate the utility of
our framework on synthetic problems as well as real applications involving
molecular structure comparison problems in chemistry.Comment: 27 pages, 7 figure
C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework
Sparse modeling is a powerful framework for data analysis and processing.
Traditionally, encoding in this framework is performed by solving an
L1-regularized linear regression problem, commonly referred to as Lasso or
Basis Pursuit. In this work we combine the sparsity-inducing property of the
Lasso model at the individual feature level, with the block-sparsity property
of the Group Lasso model, where sparse groups of features are jointly encoded,
obtaining a sparsity pattern hierarchically structured. This results in the
Hierarchical Lasso (HiLasso), which shows important practical modeling
advantages. We then extend this approach to the collaborative case, where a set
of simultaneously coded signals share the same sparsity pattern at the higher
(group) level, but not necessarily at the lower (inside the group) level,
obtaining the collaborative HiLasso model (C-HiLasso). Such signals then share
the same active groups, or classes, but not necessarily the same active set.
This model is very well suited for applications such as source identification
and separation. An efficient optimization procedure, which guarantees
convergence to the global optimum, is developed for these new models. The
underlying presentation of the new framework and optimization approach is
complemented with experimental examples and theoretical results regarding
recovery guarantees for the proposed models
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